Abstract

A new family of Nyquist pulses for coherent optical single carrier systems is introduced and is shown to increase the nonlinearity tolerance of dual-polarization (DP)-QPSK and DP-16-QAM systems. Numerical investigations for a single-channel 28 Gbaud DP-16-QAM long-haul system without optical dispersion compensation indicate that the proposed pulse can increase the reach distance by 26% and 19%, for roll-off factors of 1 and 2, respectively. In multi-channel transmissions and for a roll-off factor of 1, a reach distance increase of 20% is reported. Experimental results for DP-QPSK and DP-16-QAM systems at 10 Gbaud confirm the superior nonlinearity tolerance of the proposed pulse.

© 2012 OSA

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2011 (2)

2010 (6)

X. Zhou, X. Chen, W. Zhou, Y. Fan, H. Zhu, and Z. Li, “All-digital timing recovery and adaptive equalization for 112 Gbit/s POLMUX-NRZ-DQPSK optical coherent receivers,” J. Opt. Commun. Netw. 2(11), 984–990 (2010).
[CrossRef]

M. Nakazawa, S. Okamoto, T. Omiya, K. Kasai, and M. Yoshida, “256-QAM (64 Gb/s) coherent optical transmission over 160 km with an optical bandwidth of 5.4 GHz,” IEEE Photon. Technol. Lett. 22(3), 185–187 (2010).
[CrossRef]

B. Châtelain, C. Laperle, D. Krause, K. Roberts, M. Chagnon, X. Xu, A. Borowiec, F. Gagnon, J. C. Cartledge, and D. V. Plant, “SPM-tolerant pulse shaping for 40- and 100-Gb/s dual-polarization QPSK systems,” IEEE Photon. Technol. Lett. 22(22), 1641–1643 (2010).

C. Behrens, R. I. Killey, S. J. Savory, M. Chen, and P. Bayvel, “Nonlinear Distortion in Transmission of Higher Order Modulation Formats,” IEEE Photon. Technol. Lett. 22(15), 1111–1113 (2010).
[CrossRef]

E. Torrengo, S. Makovejs, D. S. Millar, I. Fatadin, R. I. Killey, S. J. Savory, and P. Bayvel, “Influence of pulse shape in 112-Gbit/s WDM PDM-QPSK transmission,” IEEE Photon. Technol. Lett. 22(23), 1714–1716 (2010).
[CrossRef]

A. Leven, N. Kaneda, and S. Corteselli, “Real-time implementation of digital signal processing for coherent optical digital communication systems,” IEEE J. Sel. Top. Quantum Eelectron. 16(5), 1227–1234 (2010).
[CrossRef]

2009 (3)

2008 (1)

B. Farhang-Boroujeny, “A square-root Nyquist (M) filter design for digital communication systems,” IEEE Trans. Signal Process. 56(5), 2127–2132 (2008).
[CrossRef]

2004 (1)

A. Assalini and A. M. Tonello, “Improved Nyquist pulses,” IEEE Commun. Lett. 8(2), 87–89 (2004).
[CrossRef]

1998 (1)

B. Farhang-Boroujeny and G. Mathew, “Nyquist filters with robust performance against timing jitter,” IEEE Trans. Signal Process. 46(12), 3427–3431 (1998).
[CrossRef]

1994 (1)

A. N. D’Andrea, U. Mengali, and R. Reggiannini, “The modified Cramer-Rao bound and its application to synchronization problems,” IEEE Trans. Commun. 42(234), 1391–1399 (1994).
[CrossRef]

1993 (1)

A. N. D’Andrea and M. Luise, “Design and analysis of a jitter-free clock recovery scheme for QAM systems,” IEEE Trans. Commun. 41(9), 1296–1299 (1993).
[CrossRef]

1986 (1)

F. M. Gardner, “A BPSK/QPSK timing-error detector for sampled receivers,” IEEE Trans. Commun. 34(5), 423–429 (1986).
[CrossRef]

1983 (1)

A. J. Viterbi and A. M. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with applications to burst digital transmission,” IEEE Trans. Inf. Theory 29(4), 543–551 (1983).
[CrossRef]

1980 (1)

D. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun. 28(11), 1867–1875 (1980).
[CrossRef]

1978 (1)

M. J. D. Powell, “A fast algorithm for nonlinearly constrained optimization calculations,” Lect. Notes Math. 630, 144–157 (1978).
[CrossRef]

1928 (1)

H. Nyquist, “Certain topics in telegraph transmission theory,” AIEE Trans. 47(2), 617–644 (1928).

Assalini, A.

A. Assalini and A. M. Tonello, “Improved Nyquist pulses,” IEEE Commun. Lett. 8(2), 87–89 (2004).
[CrossRef]

Awadalla, A.

Bayvel, P.

C. Behrens, S. Makovejs, R. I. Killey, S. J. Savory, M. Chen, and P. Bayvel, “Pulse-shaping versus digital backpropagation in 224Gbit/s PDM-16QAM transmission,” Opt. Express 19(14), 12879–12884 (2011).
[CrossRef] [PubMed]

E. Torrengo, S. Makovejs, D. S. Millar, I. Fatadin, R. I. Killey, S. J. Savory, and P. Bayvel, “Influence of pulse shape in 112-Gbit/s WDM PDM-QPSK transmission,” IEEE Photon. Technol. Lett. 22(23), 1714–1716 (2010).
[CrossRef]

C. Behrens, R. I. Killey, S. J. Savory, M. Chen, and P. Bayvel, “Nonlinear Distortion in Transmission of Higher Order Modulation Formats,” IEEE Photon. Technol. Lett. 22(15), 1111–1113 (2010).
[CrossRef]

Behrens, C.

C. Behrens, S. Makovejs, R. I. Killey, S. J. Savory, M. Chen, and P. Bayvel, “Pulse-shaping versus digital backpropagation in 224Gbit/s PDM-16QAM transmission,” Opt. Express 19(14), 12879–12884 (2011).
[CrossRef] [PubMed]

C. Behrens, R. I. Killey, S. J. Savory, M. Chen, and P. Bayvel, “Nonlinear Distortion in Transmission of Higher Order Modulation Formats,” IEEE Photon. Technol. Lett. 22(15), 1111–1113 (2010).
[CrossRef]

Borowiec, A.

K. Roberts, A. Borowiec, and C. Laperle, “Technologies for optical systems beyond 100G,” Opt. Fiber Technol. 17(5), 387–394 (2011).
[CrossRef]

B. Châtelain, C. Laperle, D. Krause, K. Roberts, M. Chagnon, X. Xu, A. Borowiec, F. Gagnon, J. C. Cartledge, and D. V. Plant, “SPM-tolerant pulse shaping for 40- and 100-Gb/s dual-polarization QPSK systems,” IEEE Photon. Technol. Lett. 22(22), 1641–1643 (2010).

Cartledge, J. C.

B. Châtelain, C. Laperle, D. Krause, K. Roberts, M. Chagnon, X. Xu, A. Borowiec, F. Gagnon, J. C. Cartledge, and D. V. Plant, “SPM-tolerant pulse shaping for 40- and 100-Gb/s dual-polarization QPSK systems,” IEEE Photon. Technol. Lett. 22(22), 1641–1643 (2010).

Y. Jiang, X. Tang, J. C. Cartledge, and K. Roberts, “Electronic Pre-Compensation of Narrow Optical Filtering for OOK, DPSK and DQPSK Modulation Formats,” J. Lightwave Technol. 27(16), 3689–3698 (2009).
[CrossRef]

Chagnon, M.

B. Châtelain, C. Laperle, D. Krause, K. Roberts, M. Chagnon, X. Xu, A. Borowiec, F. Gagnon, J. C. Cartledge, and D. V. Plant, “SPM-tolerant pulse shaping for 40- and 100-Gb/s dual-polarization QPSK systems,” IEEE Photon. Technol. Lett. 22(22), 1641–1643 (2010).

Châtelain, B.

B. Châtelain, C. Laperle, D. Krause, K. Roberts, M. Chagnon, X. Xu, A. Borowiec, F. Gagnon, J. C. Cartledge, and D. V. Plant, “SPM-tolerant pulse shaping for 40- and 100-Gb/s dual-polarization QPSK systems,” IEEE Photon. Technol. Lett. 22(22), 1641–1643 (2010).

Chen, M.

C. Behrens, S. Makovejs, R. I. Killey, S. J. Savory, M. Chen, and P. Bayvel, “Pulse-shaping versus digital backpropagation in 224Gbit/s PDM-16QAM transmission,” Opt. Express 19(14), 12879–12884 (2011).
[CrossRef] [PubMed]

C. Behrens, R. I. Killey, S. J. Savory, M. Chen, and P. Bayvel, “Nonlinear Distortion in Transmission of Higher Order Modulation Formats,” IEEE Photon. Technol. Lett. 22(15), 1111–1113 (2010).
[CrossRef]

Chen, X.

Corteselli, S.

A. Leven, N. Kaneda, and S. Corteselli, “Real-time implementation of digital signal processing for coherent optical digital communication systems,” IEEE J. Sel. Top. Quantum Eelectron. 16(5), 1227–1234 (2010).
[CrossRef]

D’Andrea, A. N.

A. N. D’Andrea, U. Mengali, and R. Reggiannini, “The modified Cramer-Rao bound and its application to synchronization problems,” IEEE Trans. Commun. 42(234), 1391–1399 (1994).
[CrossRef]

A. N. D’Andrea and M. Luise, “Design and analysis of a jitter-free clock recovery scheme for QAM systems,” IEEE Trans. Commun. 41(9), 1296–1299 (1993).
[CrossRef]

Fan, Y.

Farhang-Boroujeny, B.

B. Farhang-Boroujeny, “A square-root Nyquist (M) filter design for digital communication systems,” IEEE Trans. Signal Process. 56(5), 2127–2132 (2008).
[CrossRef]

B. Farhang-Boroujeny and G. Mathew, “Nyquist filters with robust performance against timing jitter,” IEEE Trans. Signal Process. 46(12), 3427–3431 (1998).
[CrossRef]

Fatadin, I.

E. Torrengo, S. Makovejs, D. S. Millar, I. Fatadin, R. I. Killey, S. J. Savory, and P. Bayvel, “Influence of pulse shape in 112-Gbit/s WDM PDM-QPSK transmission,” IEEE Photon. Technol. Lett. 22(23), 1714–1716 (2010).
[CrossRef]

Gagnon, F.

B. Châtelain, C. Laperle, D. Krause, K. Roberts, M. Chagnon, X. Xu, A. Borowiec, F. Gagnon, J. C. Cartledge, and D. V. Plant, “SPM-tolerant pulse shaping for 40- and 100-Gb/s dual-polarization QPSK systems,” IEEE Photon. Technol. Lett. 22(22), 1641–1643 (2010).

Gardner, F. M.

F. M. Gardner, “A BPSK/QPSK timing-error detector for sampled receivers,” IEEE Trans. Commun. 34(5), 423–429 (1986).
[CrossRef]

Godard, D.

D. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun. 28(11), 1867–1875 (1980).
[CrossRef]

Jiang, Y.

Kaneda, N.

A. Leven, N. Kaneda, and S. Corteselli, “Real-time implementation of digital signal processing for coherent optical digital communication systems,” IEEE J. Sel. Top. Quantum Eelectron. 16(5), 1227–1234 (2010).
[CrossRef]

Kasai, K.

M. Nakazawa, S. Okamoto, T. Omiya, K. Kasai, and M. Yoshida, “256-QAM (64 Gb/s) coherent optical transmission over 160 km with an optical bandwidth of 5.4 GHz,” IEEE Photon. Technol. Lett. 22(3), 185–187 (2010).
[CrossRef]

Killey, R. I.

C. Behrens, S. Makovejs, R. I. Killey, S. J. Savory, M. Chen, and P. Bayvel, “Pulse-shaping versus digital backpropagation in 224Gbit/s PDM-16QAM transmission,” Opt. Express 19(14), 12879–12884 (2011).
[CrossRef] [PubMed]

E. Torrengo, S. Makovejs, D. S. Millar, I. Fatadin, R. I. Killey, S. J. Savory, and P. Bayvel, “Influence of pulse shape in 112-Gbit/s WDM PDM-QPSK transmission,” IEEE Photon. Technol. Lett. 22(23), 1714–1716 (2010).
[CrossRef]

C. Behrens, R. I. Killey, S. J. Savory, M. Chen, and P. Bayvel, “Nonlinear Distortion in Transmission of Higher Order Modulation Formats,” IEEE Photon. Technol. Lett. 22(15), 1111–1113 (2010).
[CrossRef]

Krause, D.

B. Châtelain, C. Laperle, D. Krause, K. Roberts, M. Chagnon, X. Xu, A. Borowiec, F. Gagnon, J. C. Cartledge, and D. V. Plant, “SPM-tolerant pulse shaping for 40- and 100-Gb/s dual-polarization QPSK systems,” IEEE Photon. Technol. Lett. 22(22), 1641–1643 (2010).

Krause, D. J.

Laperle, C.

K. Roberts, A. Borowiec, and C. Laperle, “Technologies for optical systems beyond 100G,” Opt. Fiber Technol. 17(5), 387–394 (2011).
[CrossRef]

B. Châtelain, C. Laperle, D. Krause, K. Roberts, M. Chagnon, X. Xu, A. Borowiec, F. Gagnon, J. C. Cartledge, and D. V. Plant, “SPM-tolerant pulse shaping for 40- and 100-Gb/s dual-polarization QPSK systems,” IEEE Photon. Technol. Lett. 22(22), 1641–1643 (2010).

K. Roberts, M. O’Sullivan, K.-T. Wu, H. Sun, A. Awadalla, D. J. Krause, and C. Laperle, “Performance of dual-polarization QPSK for optical transport systems,” J. Lightwave Technol. 27(16), 3546–3559 (2009).
[CrossRef]

Leven, A.

A. Leven, N. Kaneda, and S. Corteselli, “Real-time implementation of digital signal processing for coherent optical digital communication systems,” IEEE J. Sel. Top. Quantum Eelectron. 16(5), 1227–1234 (2010).
[CrossRef]

Li, Z.

Luise, M.

A. N. D’Andrea and M. Luise, “Design and analysis of a jitter-free clock recovery scheme for QAM systems,” IEEE Trans. Commun. 41(9), 1296–1299 (1993).
[CrossRef]

Makovejs, S.

C. Behrens, S. Makovejs, R. I. Killey, S. J. Savory, M. Chen, and P. Bayvel, “Pulse-shaping versus digital backpropagation in 224Gbit/s PDM-16QAM transmission,” Opt. Express 19(14), 12879–12884 (2011).
[CrossRef] [PubMed]

E. Torrengo, S. Makovejs, D. S. Millar, I. Fatadin, R. I. Killey, S. J. Savory, and P. Bayvel, “Influence of pulse shape in 112-Gbit/s WDM PDM-QPSK transmission,” IEEE Photon. Technol. Lett. 22(23), 1714–1716 (2010).
[CrossRef]

Mathew, G.

B. Farhang-Boroujeny and G. Mathew, “Nyquist filters with robust performance against timing jitter,” IEEE Trans. Signal Process. 46(12), 3427–3431 (1998).
[CrossRef]

Mengali, U.

A. N. D’Andrea, U. Mengali, and R. Reggiannini, “The modified Cramer-Rao bound and its application to synchronization problems,” IEEE Trans. Commun. 42(234), 1391–1399 (1994).
[CrossRef]

Millar, D. S.

E. Torrengo, S. Makovejs, D. S. Millar, I. Fatadin, R. I. Killey, S. J. Savory, and P. Bayvel, “Influence of pulse shape in 112-Gbit/s WDM PDM-QPSK transmission,” IEEE Photon. Technol. Lett. 22(23), 1714–1716 (2010).
[CrossRef]

Nakazawa, M.

M. Nakazawa, S. Okamoto, T. Omiya, K. Kasai, and M. Yoshida, “256-QAM (64 Gb/s) coherent optical transmission over 160 km with an optical bandwidth of 5.4 GHz,” IEEE Photon. Technol. Lett. 22(3), 185–187 (2010).
[CrossRef]

Nyquist, H.

H. Nyquist, “Certain topics in telegraph transmission theory,” AIEE Trans. 47(2), 617–644 (1928).

O’Sullivan, M.

Okamoto, S.

M. Nakazawa, S. Okamoto, T. Omiya, K. Kasai, and M. Yoshida, “256-QAM (64 Gb/s) coherent optical transmission over 160 km with an optical bandwidth of 5.4 GHz,” IEEE Photon. Technol. Lett. 22(3), 185–187 (2010).
[CrossRef]

Omiya, T.

M. Nakazawa, S. Okamoto, T. Omiya, K. Kasai, and M. Yoshida, “256-QAM (64 Gb/s) coherent optical transmission over 160 km with an optical bandwidth of 5.4 GHz,” IEEE Photon. Technol. Lett. 22(3), 185–187 (2010).
[CrossRef]

Plant, D. V.

B. Châtelain, C. Laperle, D. Krause, K. Roberts, M. Chagnon, X. Xu, A. Borowiec, F. Gagnon, J. C. Cartledge, and D. V. Plant, “SPM-tolerant pulse shaping for 40- and 100-Gb/s dual-polarization QPSK systems,” IEEE Photon. Technol. Lett. 22(22), 1641–1643 (2010).

Powell, M. J. D.

M. J. D. Powell, “A fast algorithm for nonlinearly constrained optimization calculations,” Lect. Notes Math. 630, 144–157 (1978).
[CrossRef]

Reggiannini, R.

A. N. D’Andrea, U. Mengali, and R. Reggiannini, “The modified Cramer-Rao bound and its application to synchronization problems,” IEEE Trans. Commun. 42(234), 1391–1399 (1994).
[CrossRef]

Roberts, K.

K. Roberts, A. Borowiec, and C. Laperle, “Technologies for optical systems beyond 100G,” Opt. Fiber Technol. 17(5), 387–394 (2011).
[CrossRef]

B. Châtelain, C. Laperle, D. Krause, K. Roberts, M. Chagnon, X. Xu, A. Borowiec, F. Gagnon, J. C. Cartledge, and D. V. Plant, “SPM-tolerant pulse shaping for 40- and 100-Gb/s dual-polarization QPSK systems,” IEEE Photon. Technol. Lett. 22(22), 1641–1643 (2010).

K. Roberts, M. O’Sullivan, K.-T. Wu, H. Sun, A. Awadalla, D. J. Krause, and C. Laperle, “Performance of dual-polarization QPSK for optical transport systems,” J. Lightwave Technol. 27(16), 3546–3559 (2009).
[CrossRef]

Y. Jiang, X. Tang, J. C. Cartledge, and K. Roberts, “Electronic Pre-Compensation of Narrow Optical Filtering for OOK, DPSK and DQPSK Modulation Formats,” J. Lightwave Technol. 27(16), 3689–3698 (2009).
[CrossRef]

Savory, S. J.

C. Behrens, S. Makovejs, R. I. Killey, S. J. Savory, M. Chen, and P. Bayvel, “Pulse-shaping versus digital backpropagation in 224Gbit/s PDM-16QAM transmission,” Opt. Express 19(14), 12879–12884 (2011).
[CrossRef] [PubMed]

E. Torrengo, S. Makovejs, D. S. Millar, I. Fatadin, R. I. Killey, S. J. Savory, and P. Bayvel, “Influence of pulse shape in 112-Gbit/s WDM PDM-QPSK transmission,” IEEE Photon. Technol. Lett. 22(23), 1714–1716 (2010).
[CrossRef]

C. Behrens, R. I. Killey, S. J. Savory, M. Chen, and P. Bayvel, “Nonlinear Distortion in Transmission of Higher Order Modulation Formats,” IEEE Photon. Technol. Lett. 22(15), 1111–1113 (2010).
[CrossRef]

Sun, H.

Tang, X.

Tonello, A. M.

A. Assalini and A. M. Tonello, “Improved Nyquist pulses,” IEEE Commun. Lett. 8(2), 87–89 (2004).
[CrossRef]

Torrengo, E.

E. Torrengo, S. Makovejs, D. S. Millar, I. Fatadin, R. I. Killey, S. J. Savory, and P. Bayvel, “Influence of pulse shape in 112-Gbit/s WDM PDM-QPSK transmission,” IEEE Photon. Technol. Lett. 22(23), 1714–1716 (2010).
[CrossRef]

Viterbi, A. J.

A. J. Viterbi and A. M. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with applications to burst digital transmission,” IEEE Trans. Inf. Theory 29(4), 543–551 (1983).
[CrossRef]

Viterbi, A. M.

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Figures (19)

Fig. 1
Fig. 1

Representation in the frequency domain of Nyquist's first criterion. T = 1.

Fig. 2
Fig. 2

Frequency characteristics of the RRC (solid line) and proposed (dashed line) pulses for excess bandwidths of 50% (a), 100% (b), 150% (c), and 200% (d). T = 1.

Fig. 3
Fig. 3

Time characteristics of the RRC (solid line) and proposed (dashed line) pulses for excess bandwidths of 50% (a), 100% (b), 150% (c), and 200% (d). T = 1.

Fig. 4
Fig. 4

28 Gbaud DP-16-QAM simulation setup.

Fig. 5
Fig. 5

Spectrum of 28 Gbaud 16-QAM signals using the RRC, NRZ, and proposed pulses for α = 1 (a) and α = 1.4 (b).

Fig. 6
Fig. 6

Launch power against achievable reach distance for the NRZ, RRC and proposed pulses in single-channel transmission for α = 1 (a) and α = 1.4 (b).

Fig. 7
Fig. 7

Maximum reach against roll-off factor (α) for the RRC and proposed pulses, single-channel transmission.

Fig. 8
Fig. 8

Energy ratio against the roll-off factor for the RRC and proposed pulses.

Fig. 9
Fig. 9

Maximum reach against the roll-off factor for the RRC and proposed pulses for a channel spacing of 50 GHz (a) and 100 GHz (b), rectangular Mux/Demux response, multi-channel transmission.

Fig. 10
Fig. 10

Maximum reach against channel spacing for the RRC and proposed pulses using α = 0.5 (a), α = 1.0 (b) and α = 1.5 (c), Gaussian Mux/Demux response, multi-channel transmission.

Fig. 11
Fig. 11

Frequency response of the proposed pulse and its truncated versions for a roll-off factor of 0.5, T = 1, N = 16.

Fig. 12
Fig. 12

Eye diagrams for 16-QAM modulation with a roll-off factor of 0.5 using the RRC pulse with rectangular windowing (a) the proposed pulse with rectangular windowing (b) and the proposed pulse with optimized truncation (c), T = 1, N = 16, and M = 64.

Fig. 13
Fig. 13

Maximum reach against DAC resolution using the RRC and proposed pulses for excess bandwidths of 50% (a), 100% (b) and 150% (c).

Fig. 14
Fig. 14

Simulation setup for jitter sensitivity analysis.

Fig. 15
Fig. 15

Theoretical 16-QAM eye diagrams taken after the pulse shaper filter (a), after the matched filter (b), and after the prefilter (c), when using the RRC pulse. Theoretical 16-QAM eye diagrams taken after the pulse shaper (d), after the matched filter (e), and after the prefilter (f), when using the proposed pulse. T = 1, N = 16, M = 64 and α = 1.

Fig. 16
Fig. 16

Normalized jitter variance vs Eb/N0 for a 16-QAM system, α = 0.5 (a), α = 1 (b).

Fig. 17
Fig. 17

10 Gbaud DP-QPSK and DP-16-QAM experimental setup.

Fig. 18
Fig. 18

Measured BER vs OSNR, 10 Gbaud DP-QPSK, α = 1, back-to-back (a), 1200 km transmission at 2 dBm of launch power (b), 1200 km transmission at 4 dBm of launch power (c). Noise bandwidth = 0.1 nm.

Fig. 19
Fig. 19

Measured BER vs OSNR, 10 Gbaud DP-16-QAM, α = 1, back-to-back (a), 1200 km transmission at –2 dBm of launch power (b), 1200 km transmission at 0 dBm of launch power (c). Noise bandwidth = 0.1 nm.

Tables (1)

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Table 1 Maximum Reach of Truncated Pulses

Equations (16)

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p( kT )={ 1,k=0 0,k0 ,k=0,±1,±2,...
k= P( fk/T )= T.
P( f )={ T| f |<1/ 2T 0| f |1/ 2T .
P( f )={ T,                                              0| f |< ( 1α ) / 2T T 2 ( 1sin[ πT α ( | f | 1 2T ) ] ),   ( 1α ) 2T | f |< ( 1+α ) 2T 0,                                                | f | ( 1+α ) / 2T .
P( f ) ={ T    0| f |< ( 1α ) / 2T T/2 ( 1α ) / 2T | f |< ( 1+α ) / 2T 0   | f | ( 1+α ) / 2T .
P( f ) ={ T/3 0| f |< ( α1 ) / 2T T/2 ( α1 ) / 2T | f |< ( 3α ) / 2T T/3 ( 3α ) / 2T | f |< ( 1+α ) / 2T 0| f | ( 1+α ) / 2T .
p( t ) =( 1 1/2 )( 1α )sinc( ( 1α )πt )+ 1/2 ( 1+α )sinc( ( 1+α )πt ).
p( t ) = 1/3 ( 1+α )sinc( ( 1+α )πt )+ ( 1/2 1/3 ) [ ( 3α )sinc( ( 3α )πt )+ ( α1 )sinc( ( α1 )πt ) ].
E R = 1/ ( 2T ) | P( f ) | 2 df 0 | P( f ) | 2 df .
maximizeβ=μ s 2 ( 0 ) n= MT /2 MT /2 s 2 ( n ) +γ f=0 ( α+1 ) / 2T S 2 ( f ) f=0 N/ 2T S 2 ( f ) ,
r( kT )={ 1,k=0 0,k0 ,k=0,±1,±2,... ±M /2 .
s( n )=s( n ).
g( kT+T/2 )=0
g( kT )0,
h p ( t )=cos( 2πt/T )p( t ).
MCRB= B L T E b / N 0 log 2 ( M QAM ) T 2 4 π 2 f 2 P( f )df ,

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